60.4.8 problem 1456

Internal problem ID [11431]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 3, linear third order
Problem number : 1456
Date solved : Sunday, March 30, 2025 at 08:22:06 PM
CAS classification : [[_3rd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime \prime }+x^{2 c -2} y^{\prime }+\left (c -1\right ) x^{2 c -3} y&=0 \end{align*}

✓ Maple. Time used: 0.027 (sec). Leaf size: 73

ode:=diff(diff(diff(y(x),x),x),x)+x^(2*c-2)*diff(y(x),x)+(c-1)*x^(2*c-3)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = x \left (c_1 \operatorname {BesselJ}\left (\frac {1}{2 c}, \frac {x^{c}}{2 c}\right )^{2}+c_2 \operatorname {BesselY}\left (\frac {1}{2 c}, \frac {x^{c}}{2 c}\right )^{2}+c_3 \operatorname {BesselJ}\left (\frac {1}{2 c}, \frac {x^{c}}{2 c}\right ) \operatorname {BesselY}\left (\frac {1}{2 c}, \frac {x^{c}}{2 c}\right )\right ) \]

✓ Mathematica. Time used: 0.017 (sec). Leaf size: 183

ode=(-1 + c)*x^(-3 + 2*c)*y[x] + x^(-2 + 2*c)*D[y[x],x] + Derivative[3][y][x] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to c_1 \, _1F_2\left (\frac {1}{2}-\frac {1}{2 c};1-\frac {1}{c},1-\frac {1}{2 c};-\frac {x^{2 c}}{4 c^2}\right )+4^{-1/c} c^{-2/c} c_3 \left (x^{2 c}\right )^{\frac {1}{c}} \, _1F_2\left (\frac {1}{2}+\frac {1}{2 c};1+\frac {1}{2 c},1+\frac {1}{c};-\frac {x^{2 c}}{4 c^2}\right )+2^{-1/c} c^{-1/c} c_2 \left (x^{2 c}\right )^{\left .\frac {1}{2}\right /c} \, _1F_2\left (\frac {1}{2};1-\frac {1}{2 c},1+\frac {1}{2 c};-\frac {x^{2 c}}{4 c^2}\right ) \]

✗ Sympy

from sympy import * 
x = symbols("x") 
c = symbols("c") 
y = Function("y") 
ode = Eq(x**(2*c - 3)*(c - 1)*y(x) + x**(2*c - 2)*Derivative(y(x), x) + Derivative(y(x), (x, 3)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - (-c*y(x) - x**(3 - 2*c)*Derivative(y(x), (x, 3)) + y(x))/x cannot be solved by the factorable group method